Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122515] Re: Integral points on elliptic curves
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 30 Oct 2011 04:25:19 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j80q8p\$adb\$1@smc.vnet.net> <j862sg\$5v7\$1@smc.vnet.net> <201110262140.RAA00128@smc.vnet.net> <j8dthn\$kga\$1@smc.vnet.net> <201110291112.HAA05405@smc.vnet.net>

```On 29 Oct 2011, at 13:12, Costa Bravo wrote:

> Andrzej Kozlowski wrote:
>
>>> In[60]:= k = 1641843;
>>> AbsoluteTiming[y0 = Ceiling[k^(1/3)];
>>>  Do[If[FractionalPart[Sqrt[y^3 - k]] == 0,Print[Sqrt[y^3 - k], " ", y]], {y, y0, 10000}]]
>>>
>>>   468  123
>>> 11754  519
>>>
>>> Out[61]= {3.4375000, Null}
>>>
>>> --
>>>  Costa
>>>
>>
>> This is hardly impressive when compared with:
>>
>>  Solve[y^3 - x^2 == 1641843&&  0<  x&&  0<  y<  10^3, {x, y},
>>   Integers] // Timing
>>
>> {0.383947,{{x->468,y->123},{x->11754,y->519}}}
>>
>> Andrzej Kozlowski
>
> O , You have y<10^3  I have y <= 10 000 !!
>
>    Solve[y^3 - x^2 == 1641843&&  0<  x&&  0<  y<  10^4, {x, y},
>    Integers] // Timing
>
>  {10.22, {{x -> 468, y -> 123}, {x -> 11754, y -> 519}}}
>
> This is hardly impressive ;)
>
> If in your algorithm, we take y< 2*10^4 -> Solve::svars !
>
> --
>  Costa
>

SetSystemOptions[
"ReduceOptions" -> {"ExhaustiveSearchMaxPoints" -> {1000, 10^6}}];

Solve[y^3 - x^2 == 1641843 && 0 < x && 0 < y < 10^6, {x, y},
Integers] // Timing

{1050.14,{{x->468,y->123},{x->11754,y->519}}}

Andrzej Kozlowski

```

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